fitting the viking lander surface pressure cycle with a ... · fitting the viking lander surface...

Fitting the Viking lander surface pressure cycle with a Mars General

Circulation Model

Xin Guo,1 W. Gregory Lawson,1 Mark I. Richardson,1,2 and Anthony Toigo3

Received 14 November 2008; revised 18 March 2009; accepted 21 May 2009; published 30 July 2009.

[1] We present a systematic attempt to fit the Viking lander surface pressure cycle using aMars General Circulation Model, MarsWRF. Following the earlier study by Wood andPaige (1992) using a one-dimensional model, high-precision fitting was achieved bytuning five time-independent parameters: the albedo and emissivity of the seasonal capsof the two hemispheres and the total CO2 inventory in the atmosphere frost system. Weused a linear iterative method to derive the best fit parameters: albedo of the northerncap = 0.795, emissivity of the northern cap = 0.485, albedo of the southern cap = 0.461,emissivity of the southern cap = 0.785, and total CO2 mass = 2.83 � 1016 kg. If theseparameters are used in MarsWRF, the smoothed surface pressure residual at the VL1 site isalways smaller than several Pascal through a year. As in other similar studies, the best fitparameters do not match well with the current estimation of the seasonal cap radiativeproperties, suggesting that important physics contributing to the energy balance notexplicitly included in MarsWRF have been effectively aliased into the derived parameters.One such effect is likely the variation of thermal conductivity with depth in the regolithdue to the presence of water ice. Including such a parameterization in the fittingprocess improves the reasonableness of the best fit cap properties, mostly improvingthe emissivities. The conductivities required in the north to provide the best fit are higherthan those required in the south. A completely physically reasonable set of fit parameterscould still not be attained. Like all prior published GCM simulations, none of thecases considered are capable of predicting a residual southern CO2 cap.

Citation: Guo, X., W. G. Lawson, M. I. Richardson, and A. Toigo (2009), Fitting the Viking lander surface pressure cycle with a

Mars General Circulation Model, J. Geophys. Res., 114, E07006, doi:10.1029/2008JE003302.

1. Introduction

[2] In every atmospheric General Circulation Model(GCM), one of the first and foremost issues to consider isthe total mass of the atmosphere. Its value affects all subse-quent calculations performed in the GCM, including dynam-ics, radiative transfer, tracer and energy transport, chemicalreactions, etc. When a component of the atmosphere isvolatile (changes state for commonly encountered environ-mental conditions), the surface pressure may be affected. Forthe Earth, where the abundance of water vapor is modestlyhigh, many GCMs include the varying contribution of watervapor partial pressure when calculating the full surfacepressure. For planets where their major atmospheric constit-uents condense (such as Mars, Triton, and Pluto), consider-ation of the role of phase change in varying the surfacepressure is important.

[3] Carbon dioxide (CO2) is the leading gaseous specieson Mars, comprising 95% of the atmosphere [Owen et al.,1977]. Throughout a Martian year, up to 30% of the totalCO2 condenses onto the surface in winter polar regions [Kellyet al., 2006]. The condensation of CO2 occurs because theMartian atmosphere is sufficiently thin that transport ofsensible heat is unable to sustain the winter polar atmosphere.As a result, the temperatures fall until the frost point of CO2

(also loosely referred to as ‘‘condensation point,’’ althoughcondensation usually refers to the phase change from gas toliquid) is reached, after which thermal infrared losses arebuffered by latent heating.[4] The Viking landers (VLs) provide the only interannual

records of surface pressure on Mars. These records containvariability on several different time scales. Variations on timescales of seconds, hours, and days are associated withboundary layer turbulence, the thermal tides, and large-scaleweather systems, for example. However, when the highfrequency component is ignored, the surface pressure varia-tion at a specific location on Mars (Figure 1) is largelydetermined by the bulk mass of atmosphere, though smallcontributions due to standing systems over long periods likethe condensation flow and the intertropical lows and sub-tropical highs are also present. Therefore, to some significantdegree of precision, the VL pressure records can be taken astopographically shifted records of the bulk atmospheric mass

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, E07006, doi:10.1029/2008JE003302, 2009

1Division of Geological and Planetary Sciences, California Institute ofTechnology, Pasadena, California, USA.

2Ashima Research, Pasadena, California, USA.3Center for Radiophysics and Space Research, Cornell University,

Ithaca, New York, USA.

Copyright 2009 by the American Geophysical Union.0148-0227/09/2008JE003302$09.00

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[Hess et al., 1977]. Fitting the observed VL pressure cyclealso establishes a baseline for climate studies, both short-termand long-term. As shown in Figure 1, the dominant long-period signal is the very repetitive annual cycle that can beidentified with the atmosphere freezing out to form theseasonal polar caps, followed by the caps’ decay due tosublimation back to the atmosphere [Hourdin et al., 1993].Models are often tuned to match this annual cycle, thoughsmall errors in amplitude and (especially) phase are oftentolerated, depending on the interests of a particular study[Forget et al., 1998; Haberle et al., 2008; Hourdin et al.,1993; Pollack et al., 1993; Richardson and Wilson, 2002].Indeed, the quality of fits in GCMs is in some sense sur-prisingly poor. For a system that should be driven primarilyby radiative, latent, and thermal conductive heating processesat the caps, it would seem that a near perfect fit should beattainable. An accurate simulation of the pressure cyclewould also be desirable for many applications where goodprediction of the surface pressure is needed as a boundarycondition, such as surface wind stress calculation, spacecraftentry-descent-landing analysis, etc. In reality, the main rea-son that ideal fits have not been attained is due to thecomputation involved: the problem reduces to one of search-ing for the best fit parameters using a relatively large numberof simulations (certainly ‘‘large’’ by standards prior to theearly 2000s).[5] There are historical attempts to ‘‘fit’’ the CO2 cycle.

Wood and Paige [1992] proved that it is possible to fit theVL data to within several Pascal using a one-dimensional(1-D) diurnal and seasonal thermal model without anyexplicit atmospheric contribution to the heat balance. The

albedos and emissivities of the northern and southernseasonal CO2 caps and total mass of CO2 in the systemwere tuned and kept constant with time. In their 1-D study,the residual between the modeled pressure in the best casescenario and the observed was as small as a few Pascal.However, their best fit parameter values differ from thecurrent estimates for the caps’ radiative properties. Ifvalues from Wood and Paige [1992] are used in a GCM,the resulting pressure curves are not unreasonable, but thelack of an atmosphere and the use of constant thermalinertia translates to offsets that yields a GCM cycle nota-bly worse than what the same parameters produces in the1-D model.[6] It is clear from 1-D modeling that tuning of the cap

properties will allow the modeled CO2 cycle to be improved.It seems the standard for Mars GCMs is to accomplish thistuning ‘‘by-eye.’’ However, the resulting parameters are notphysical compared with spacecraft observations. For exam-ple, the cap albedos have to differ dramatically between thepoles, with the northern values required to be much higherthan observed [Haberle et al., 2008; Kieffer et al., 2000;Kieffer and Titus, 2001]. One possible way to improve thereasonableness of the parameters is to include other effects,such as the thermal conductivity induced by subsurface waterice [Haberle et al., 2008]. However, to date there has not beenan attempt to systematically determine the cap propertiesneeded in a GCM to produce a seasonal pressure cycle withinthe VL instrument error. Here, we present a recipe that can beused to calibrate a GCMpressure cycle, and we provide somediscussion of what areas of uncertainty still remain in theunderstanding of the CO2 cycle on Mars.

Figure 1. (top) Pressures at the VL sites for the control case (northern seasonal cap albedo, 0.77; northernseasonal cap emissivity, 0.57; south albedo, 0.5; south emissivity, 0.8; total CO2mass index, 1.0). Grey line,MarsWRF surface pressure at VL1 location; magenta dashed line, smoothed MarsWRF surface pressure atVL1 location; black line, MarsWRF surface pressure at VL2 location; green dashed line, smoothedMarsWRF surface pressure at VL2 location; blue dashed line, smoothed VL1 observation; red dashed line,smoothed VL2 observation. (bottom) Difference between the smoothed VL measurements and MarsWRFsimulation: blue line, VL1 site; red line, VL2 site.

E07006 GUO ET AL.: FIT MARS CO2 CYCLE WITH MARSWRF

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[7] In this paper, we describe the development andexecution of a benchmark study for reproducing the observedsurface pressure cycle in a Mars GCM. We specifically usethe Martian implementation of the planetWRF model[Richardson et al., 2007]. The reason that Wood and Paige[1992] were able to generate excellent fits of the VL pressurecycle is that their 1-D model is simple and computationallycheap: hundreds of runs were made to cover the parameterspace so that a best fit can be identified. Hundreds of runs forany sophisticated GCM is generally not feasible in a reason-able amount of time, nor are such runs necessary with ourproposed method. Our technique involves the creation of anensemble of simulations that are used to understand how themodel’s representation of the VL surface pressure cyclesresponds to perturbations of different parameters. Knowingthese relationships, we use an iterative linear method to findthe best set of parameters to fit the VL data. The method canfit the VL data with a very high degree of accuracy: com-parable to, if not better than, the fit using the simpler one-dimensional model [Wood and Paige, 1992]. This approachis non model specific and is suggested as an efficient meansof finding the parameters for a best fit pressure cycle for anyGCM. After demonstrating the method for the set of param-eters chosen by Wood and Paige [1992], we proceed furtherto study how changes in soil thermal properties due to sub-surface water ice effect the model’s representation of the VLsurface pressure record.[8] The fact that Wood and Paige [1992] were able to fit

the pressure cycle using a simple model has profoundmeaning. It suggests that the physics included in the model,namely the surface heat balance at the poles, is the control-ling physics for modeling the CO2 and pressure cycles. Thenonphysicality of the cap parameters retrieved from theWoodand Paige [1992] scheme does not invalidate the approach:instead it demonstrates that there exists a range of processesthat effect the polar energy balance that are not incorporated(‘‘resolved’’) in the model and hence their effects are‘‘aliased’’ (or ‘‘bundled’’) into the estimated best fit param-eters: for example, an unrealistic albedo value may effectivelycompensate for model-missing clouds. Further progress inunderstanding the CO2 cycle, which is tantamount to under-standing the polar energy balance, is now a question ofunbundling the various processes that are aliased into theretrieved parameters. The way this can be done is by applyingmodels withmore complete sets of explicit physics. Applyingthe same fitting routines within a hierarchy of increasinglycomplex models will allow the progressive unbundling ofimportant physics affecting the polar heat balance. Wherewill it all end? At some point a set of physically plausibleradiative parameters (potentially in the form of time andspatially varying parameters defined by a physical under-standing of the ice microphysics and formation history) willsignal the completion of the unbundling process. For exam-ple, in this study, we apply a GCM to the problem studiedin 1-D by Wood and Paige [1992]. Explicit treatment ofatmospheric dynamics, and spatially varying topographyand regolith albedo and thermal properties can thus beunbundled from the parameters found by Wood and Paige[1992]. The complexity of implementing this task arrivesfrom the very large numbers of runs required to find excellentdata fits. Such a brute-force method is not available whenrunning computationally expensive GCM simulations.

Hence, much of our discussion in this paper is in thedemonstration of a method to allow data fitting with arelatively small number of runs. In principal, modifiedversions of this approach can be applied to fit various otherdata sets (e.g., temperature). In essence, it is a limitedimplementation of a prototype form of data assimilation (herewe are pursing ‘‘parameter estimation’’ while most dataassimilation attempts ‘‘state estimation’’). Obviously, anideal approach would be to simultaneously fit the surfacepressure data along with records of air temperature, dust,water ice, etc. This in fact would require a full parameter andstate data assimilation system, which is a future goal of ourresearch. That notwithstanding, the approach and schemespresented here allow some new insight into the CO2 cyclewithout having to resort to a full data assimilation system.[9] In section 2, we discuss the relevant physical processes

contributing to surface energy balance, and we present ourinitial sensitivity study for several important contributingparameters. In section 3, we introduce the iterative linearfitting method. We show the fitting results in section 4.Physical interpretation of the best fit parameters is discussedin section 5. Section 6 concludes this paper.

2. Surface Energy Balance and Sensitivity Study

2.1. Surface Energy Balance

[10] The key physics for the Martian annual CO2 cycle isthe surface energy balance. The instantaneous surface energybalance equation when CO2 frost is on the surface is

S 1� að Þcos ið Þ � esT4 þ k dT=dzþ L dm=dt ¼ 0; ð1Þ

where S is the incoming solar flux at the current Mars-Sundistance; a is the albedo, either that of bare soil or frost; i isthe solar incidence angle; e is the surface emissivity, eitherthat of bare soil or frost; s is the Stefan-Boltzmann constantfor blackbody emission; T is the surface temperature, eitherthat of bare soil or the frost temperature when frost presents;k is the thermal conductivity of the soil; dT/dz is the verticaltemperature gradient at the surface with z positive down-ward (therefore, k dT/dz is the upward conductive heat fluxat the surface); L is the latent heat of CO2 frost; dm/dt is theCO2 frost deposition/sublimation rate. The thermal con-ductivity of a material is related to its thermal inertia (I) bythe equation I2 = krc, where r is the density and c is the heatcapacity [Wood and Paige, 1992]. The product of r and cis assumed to be 1.26 � 106 J kg m�6 K�1 in MarsWRF.This is a simplified view of the problem. The GCM containsadditional terms associated with the atmospheric componentsthat contribute to this energy balance, which are usuallymuchsmaller when compared to the rest. Thus for simplicity with-out losing clarity to the readers, we will refer to this equation,which captures the majority of the physics.[11] At any given time and location, when the radiative

and sensible heating terms in equation (1) are negative, latentheating is required to balance the cooling and maintain thetemperature at the condensation point (the temperatureremains at the condensation point because any infinitesimaltemperature drop would yield a drop in the required satura-tion vapor pressure. The difference between the actualsurface pressure and the infinitesimally lower saturationvapor pressure would drive CO2 from the atmosphere onto


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the surface, liberating latent heat that would tend to bring thetemperature back up to the point where the atmosphericpressure and the saturation vapor pressure again agree). Asa result, negative net radiative and sensible heating yieldsCO2 gas conversion to ice and deposition on the surface.When net heating is positive and CO2 ice is present on thesurface, some amount of CO2 frost becomes gas until thesurface is exhausted of all its CO2 ice cover. These phaseexchange processes are usually assumed to be instantaneousin GCMs.[12] Extensive work has been done examining water ice

cloud and CO2 ice cloud formation. Modeling studies foundevidence showing their importance in the polar energybudget by influencing the surface properties, temperature,and dynamics [Colaprete et al., 2005, 2008; Richardson etal., 2002]. Because the microphysics modeling of clouds inthe Martian atmosphere is a subject under investigation, wedo not include CO2 cloud formation or a water cycle in thisstudy (these processes will await further unbundling at afuture date).[13] In selecting parameters to use for the fitting of the

CO2 cycle, we take into account what variables are going tohave a major effect on the polar energy balance and alsowhat aspects of the model are relatively well constrained.For the radiative transfer, we use our standard solar andthermal infrared radiative heating schemes and also chose toprescribe the dust opacity distribution with the Mars ClimateDatabase ‘‘MGS’’ scenario parameterization. Global soilalbedo, emissivity and surface thermal inertia are used asderived from spacecraft observations [Richardson et al.,2007]. The parameters chosen to vary for the simulation arethose associated with the seasonal caps and the bulkinventory of CO2: the albedo and the emissivity of theseasonal CO2 ice caps (potentially separate values for eachpole), the global CO2 mass (both gaseous and solid phase),and the thermal inertia of the polar regolith, which is amixture of soil and water ice. All the parameters are crucialto the energy balance and the pressure cycle. In the sectionsfollowing, we first discuss the conventional tuning param-eters, namely albedo, emissivity, and total CO2 inventory.Then we discuss the less-studied varying thermal propertyof the subsurface.

2.2. Usual Suspects: Albedo, Emissivity,and Total CO2 Inventory

[14] The albedo determines the fraction of incoming solarenergy reflected back to space. In general, increasing thealbedo decreases the absorbed energy available to the land-air system. In the Martian polar winter, as temperatures dropto the CO2 frost point and CO2 deposits on the surface, thehigh (bright) ice albedo further lowers solar heating, andmore CO2 freezes out. The magnitude of this feedback effectdepends on the value of the frost albedo. Thus, surfacepressure tends to lower as a result of an increased CO2 capalbedo, and vice versa. We note that this effect is onlyrelevant when CO2 frost is present and it is receivinginsolation (an example of this is shown in Figure 3 (top),discussed later).[15] The emissivity of the seasonal caps participates in

maintaining surface energy balance differently than albedodoes. The larger the emissivity of a CO2 frost cap, the moreenergy it releases to space. Thus a larger emissivity yields a

larger energy deficit, which in turn requires more conden-sation of CO2 to release latent heat and compensate for theenergy deficit. The net effect is that an increase in emissivityleads to a decrease in the surface pressure. Similarly toalbedo’s effect, we note that the effect of emissivity onsurface pressure is only relevant when CO2 frost is present,though it can operate throughout polar night (this is alsopresent in Figure 3 (top), discussed later). This means that theeffects from albedo changes and from emissivity changeshave different longevity and prominence: while albedo onlyacts when the seasonal caps are exposed to the sun, emissivityacts as long as the surface frost exists. Therefore, the albedo‘‘footprint’’ (the effect on the surface pressure cycle due tovariations of albedo) for a given pole is nonzero only later inits season of frost coverage, whereas the emissivity footprintis present throughout the frost coverage season.[16] The total mass of CO2 in the system influences the

surface pressure cycle in a very linear manner. Since none ofthe terms in equation (1) are very sensitive to total atmo-spheric mass, the surface frost amount changes very littlewhen total mass of CO2 is changed (this is also present inFigure 3 (top), discussed later). Of course, one should notmake very large perturbations to the CO2 inventory lest theplanet migrate to another climate regime [Mischna et al.,2000]. It should also be noted that since the model does notpredict the development of a residual CO2 ice cap (nor doesany published GCM), increasing the surface pressure is pos-sible by increasing the total CO2 inventory.

2.3. Water Ice in the Subsurface Layer

[17] In equation (1), the vertical heat flux is determinedby the temperature gradient near the surface and the soil’sthermal conductivity. Subsurface thermal structure is deter-mined by the energy input from the surface and the thermalproperty underneath. Typically, the subsurface thermal con-ductivity is assumed to be the same as that of the surface. Ithas recently been suggested that the subsurface water icewould affect the exchange of CO2 between the atmosphereand the seasonal caps greatly [Haberle et al., 2008]. Thepresence of water ice increases the thermal conductivity ofthe soil. In the summer, when the surface is not covered byfrost, more heat conducts downward compared to the situa-tion where homogenous dry soil is usually assumed. Whenthis extra amount of heat is released in the winter, less sur-face CO2 ice forms, and surface pressure increases accord-ingly. Therefore, changing the thermal inertia (equivalent tothe thermal conductivity) of the regolith changes the thermalstructure and heat flux of the soil. Eventually, it changes thesurface pressure cycle.[18] In order to investigate how subsurface water ice

affects the CO2 cycle, we modified the subsurface modelin the MarsWRF to handle vertical gradients of soil thermalconductivity. At depth where water ice becomes stable andhas significant mass, we increase the thermal conductivityof the soil. This is similar to the two layer model used byHaberle et al. [2008], in which the regolith beneath a certaindepth (8.05 cm for the northern hemisphere and 11.16 cmfor the southern hemisphere) was fully filled by a water icetable from 55� to 90� latitude (Figure 2, right); however, wedo not use fixed depths across the two hemispheres.[19] In order to parameterize the three-dimensional dis-

tribution of water ice in the subsurface, we include the water


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ice content map provided by the Mars Odyssey Gamma RaySpectrometer (GRS) [Boynton et al., 2002; Feldman et al.,2002, 2004] and the depth of the permanently stable sub-surface water ice predicted by Schorghofer and Aharonson[2005] in MarsWRF (Figure 2). Water ice is only seasonallystable at midlatitudes [Schorghofer and Aharonson, 2005],and the available water vapor in the atmosphere does notseem to be enough to fill meters of regolith [Smith, 2002].Therefore, we set the water ice table depth to be the cutoffline of the permanently stable water ice, which changes withlatitude and longitude.[20] The other parameter to decide upon is the thermal

conductivity, or equivalently the thermal inertia, of the soilmixed with water ice (hereafter, we loosely call it a waterice table even though it is not pure water ice). VikingInfrared Thermal Mapper (IRTM) observations show thatthe apparent thermal inertia of the dry Martian surfaceranges from 46 to 630 J m�3 K�1, with a global averageof 275 J m�3 K�1 [Paige, 1992]. This agrees with obser-vations from the Thermal Emission Spectrometer (TES) onboard of Mars Global Surveyor (MGS) [Putzig et al., 2005].Pure water ice has a thermal inertia of 2200 J m�3 K�1.Assuming a mixture has 80% of water ice, the thermal inertiaof the mixture can range from 100 to 1200 J m�3 K�1,depending on the thermal inertia of the host regolith. Thesenumbers provide physical constraints for the fitting param-eters in later sections. We do not assume the water contentprovided by GRS for the water ice table because that data setis not vertically well resolved. The water information fromGRS is only used for the geographical distribution.We set the

threshold for water content to 9% in order to cover thelatitudes of 55� and poleward.[21] Of course, we do not have to limit ourselves to these

numbers in the sensitivity study.We find that the VL pressurecycle is very sensitive to the thermal inertia assumed for thewater ice table (Figure 3, bottom). A change in the assumedwater ice table thermal inertia changes the VL pressure cyclesignificantly. As expected, larger thermal inertia leads tohigher surface pressure. We find that the water ice tablethermal inertia shows a very similar footprint to that of theemissivity: the increasing phase of their signals lasts from Ls210� to 360� in the northern winter and from Ls 30� up to180� in the southern winter; the decreasing phase of thethermal inertia signal ends shortly before that of the emis-sivity. However, we find that thermal inertia does not projecta change on the surface pressure cycle as linearly as emis-sivity does. For example, doubling the thermal inertia per-turbation does not necessarily double the resulting pressurechange. In addition, changing the thermal inertia modifies thephase of the pressure signal. Also of note, the sensitivity tothermal inertia evidently saturates: when we extend thethermal inertia above a certain amount, the pressure cyclefails to respond. At that point, the soil in the model containsthe maximum possible amount of heat it can. Fortunately forthe prospect of fitting the pressure cycle, with the currentdepth of soil, MarsWRF seems to reach that extreme for athermal inertia value larger than 5000 J m�3 K�1, which ismuch larger than most of the possible material in the regolithincluding pure water ice. Nonetheless, we need to be carefulwhen we perform the linear retrieval later, where we assume

Figure 2. (left) Color shows the subsurface ice content from GRS; white contours indicate the depth ofpermanently stable water ice in meters suggested by Schorghofer and Aharonson [2005]. (right) Blackcurve indicates the zonal average of the depth of permanently stable water ice table; blue curve indicatesthe ice table depth used by Haberle et al. [2008].


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the pressure cycle always responses linearly to all theperturbations.[22] The average thermal inertia of dry Martian soil has

an annual skin depth around 1 m, which is usually deeperthan the stable water ice depth. Our subsurface model inMarsWRF covers more than 7 m, deeper than the annual skindepth of pure water ice (thermal inertia of 2200 J m�3 K�1,skin depth 5 m). The stable water ice depth was calculatedassuming the soil porosity is about 70%. The pressure cycleis not very sensitive to the assumption of the underlyingsoil porosity. This is because the stable ice depth does notchange much with soil porosity near the poles, where thelow temperature allows subsurface water ice to preserveeasily.

2.4. Mathematical Representation

[23] In order to simplify the discussion and the descrip-tion of the linear fitting method, we introduce the followingmathematical notation. In the initial experiments, five param-eters in MarsWRF are tuned: the albedo and emissivity of theseasonal CO2 caps for both the north and south poles, and thetotal mass of CO2 in the system. The linear methodologydiscussed in section 3 relies on making relatively smallcorrections; hence we constructed a baseline case withparameters similar to those used by Wood and Paige[1992]. The northern cap albedo is set to 0.770, northerncap emissivity to 0.570, southern cap albedo to 0.500,southern cap emissivity to 0.800 and total CO2 mass to

2.90 � 1016 kg of CO2 (we use an index number 1.00 here-after to represent this reference total CO2 amount).We denotethe parameter vector as A, a column vector whose rows con-tain the parameters used in an experiment in the order justmentioned. For the baseline case, we write

A0 ¼ 0:770 0:570 0:500 0:800 1:00½ �T; ð2Þ

where superscript T denotes vector transpose. The valuesin this parameter vector are used in MarsWRF to yield itscorresponding baseline model atmosphere. After the modelreaches a steady state (taken to be aMartian year; note that theinitial subsurface temperature were taken from the results of aprior decadal simulation and were thus very nearly in balancefrom the outset), a Mars year of surface pressure output isdiagnosed for its predicted values of the Viking Lander 1(VL1) data record. This diagnosis requires interpolatingwithin the output to find the surface pressure at the actuallatitude, longitude, and elevation of the lander (gray line inFigure 1). It is evident that the model’s predicted pressurecycle has larger short-term variations in the second half ofthe year, and this is consistent with observations and previousGCM studies. Details of those short-term variations arebeyond the scope of this study. For the purpose of tuning theuncertain parameters in our vector, we are only interestedin the long-term trend. Hence we apply a 9-day runningaveraging to the model output’s predicted surface pressure

Figure 3. Response of MarsWRF surface pressure at VL1 location after positive perturbations.(top) Perturbation to north seasonal cap albedo (thin black line, P1), north emissivity (thin gray line, P2),south albedo (thick black dash line, P3), south emissivity (thick gray dash line, P4), and total CO2 mass(thick black line, P5). (bottom) Perturbation to the thermal inertia of the water ice table at the north polarregion by 600 J m�3 K�1 (thin black dash line, P6

e), south by 600 J m�3 K�1 (black line, P7e), both by

600 J m�3 K�1 (thick gray line, Pextrae ), both by 1000 J m�3 K�1 (thick black line), both by 2000 J m�3 K�1

(thick black dash line), and both by 3000 J m�3 K�1 (thick gray dash line). The last two curves only appearpartially because of the scale limit.


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record to remove the undesired high frequency components.This smoothed pressure cycle, X0, a column vector with669 rows (each row represents a sol) is plotted in Figure 1 asthe dashed magenta line. We write

X0 ¼ I W A0ð Þð Þ F A0ð Þ; ð3Þ

where operator I denotes all the required interpolations andsmoothing, and operator W denotes the MarsWRF modelwhen run with the parameters included in A0. We can com-bine these two operators into one effective operator, F, thatmaps a given parameter vector to a predicted smoothed VL1pressure record.[24] In order to sensibly compare our smoothed model

predicted pressure cycle to the observations, which alsoexhibit high frequency variations due to tides and baroclinicweather systems, we have used a smoothed, continuousrepresentation of the VL1 observations in non dust stormconditions. Instead of applying the same 9-day runningaverage as we have to the model output, we use the polyno-mial fit of the non dust storm VL1 pressure curve [Tillman etal., 1993]. Using a polynomial allows us to compare themodel to the data in regions where there are gaps in the VL1data record. The smoothed VL1 observations are presentedby the dashed blue line in Figure 1, with one value of VL1surface pressure per sol (669 values in a Martian year). Wefollow the same practice with the VL2 observation, shown inFigure 1 as the red dashed curve, though we do not use theVL2 observations within the linear fitting method.[25] As is evident in Figure 1, the smoothed pressure

cycle generated using the baseline parameter set (magentadashed line for VL1 and green dashed line for VL2) differsfrom the observations (blue dashed line for VL1 and reddashed line for VL2). The largest differences are in thenorthern summer (or the southern winter), where phase errorsare evident. The residual, defined as the model predictionsless the VL observations, vary from 0 to 20 Pa throughoutmost of the year, with an average of 8.2 Pa and a standarddeviation of 7.0 Pa at the VL1 site. The root mean square(RMS) of the residual at the VL1 site is 10.8 Pa, or 1.2–1.6%of the seasonal cycle.[26] To estimate the sensitivities of MarsWRF to the

parameters within A0, we make five additional MarsWRFmodel runs, one for each element in A0. Each run adds asmall perturbation to the value of one parameter in A0 whilekeeping the other values unchanged. Once completed, weapply the same interpolation and averaging to obtain thefive predicted smooth pressure cycles at the VL1 locationassociated with the five perturbed parameter vectors. Wedenote the perturbed parameter vectors asAi (i = 1, 2, 3, 4, 5),and analogously to equation (3), their resulting pressurerecords are Xi (i = 1, 2, 3, 4, 5). To be clear, the order ofthe perturbations associated with the i index follows the orderof parameters within A0: northern seasonal cap albedo,northern cap emissivity, southern cap albedo, southern capemissivity and the index number of the total CO2 mass inMarsWRF, respectively. We chose the following set ofperturbed parameter vectors Ai:

A1 ¼ 0:820 0:570 0:500 0:800 1:00½ �T; ð4Þ

A2 ¼ 0:770 0:670 0:500 0:800 1:00½ �T; ð5Þ

A3 ¼ 0:770 0:570 0:600 0:800 1:00½ �T; ð6Þ

A4 ¼ 0:770 0:570 0:500 0:900 1:00½ �T; ð7Þ

A5 ¼ 0:770 0:570 0:500 0:800 1:06½ �T: ð8Þ

These vectors are related to the baseline parameter vectorthrough

Di ¼ Ai � A0; i ¼ 1; 2; 3; 4; 5ð Þ: ð9Þ

Similarly, we define the perturbation pressure vectors

Pi ¼ X i � X 0 ¼ F Aið Þ � F A0ð Þ; i ¼ 1; 2; 3; 4; 5ð Þ: ð10Þ

While a matrix whose columns are the Di vectors is a 5 � 5diagonal matrix, a matrix whose columns are the Pi vectorsis 669 � 5 and potentially has nonzero entries everywhere.The Pi vectors are shown in Figure 3 (top), and one can seethat they are in general nonzero, except for the vectorsassociated with perturbed albedo and emissivity when theyare not contributing to the surface energy balance.[27] This same procedure for gauging the model’s sensi-

tivity to uncertain parameters can easily incorporate othercomponents. Later in this study, we extend the parametervector to include the water ice table thermal inertia. Thisrequires defining a new baseline case with parameter vector

Ae0 ¼ 0:770 0:570 0:500 0:800 1:00 1:20 1:20½ �T; ð11Þ

where the definitions of the first five elements of this vectorare the same as before, and the sixth and seventh elementscorrespond to the thermal inertia of the subsurface layer inthe northern polar region and the southern polar region, inunits of 1000 J m�3 K�1. We use the same notation for theoperators that map a given parameter vector to a predictedsmoothed pressure curve for VL1:

X e0 ¼ I W Ae

0

� �� ¼ F Ae

0

� �: ð12Þ

To be clear, in the experiments where only the first fiveparameters are varied, MarsWRF still runs with assumedvalues for the sixth and seventh parameters. However, theirvalues are not available for modifying to better fit the VL1pressure data. When the parameters for subsurface thermalinertia in the two hemispheres are available for change, wespecify two new parameter vectors:

Ae6 ¼ 0:770 0:570 0:500 0:800 1:00 1:50 1:20½ �T; ð13Þ

Ae7 ¼ 0:770 0:570 0:500 0:800 1:00 1:20 1:50½ �T: ð14Þ

These extended perturbed parameter vectors logically extendto the definitions of D6

e, D7e, P6

e and P7e via equations (9) and

(10). The corresponding perturbation pressure vectors for P6e


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and P7e are shown in Figure 3 (bottom). Also shows is one

additional perturbation pressure vector obtained from run-ning MarsWRF with

Aeextra ¼ 0:770 0:570 0:500 0:800 1:00 1:50 1:50½ �T:

This is included to demonstrate that Pextrae is roughly the

sum of P6e and P7

e, thereby demonstrating these parametersbehave fairly linearly. Note that Pextra

e is not used in thefitting method described in section 3.

3. Linear Fitting Methods

[28] We chose to fit the pressure cycle at just one of theVL sites, reserving the other as an independent resource tocheck the retrieved set of parameters. Since the VL1 pres-sure record is much less susceptible to local weather distur-bances and large global dust storms, and is more complete inturns of time coverage than the VL2 record, the VL1 data aretaken to be the preferable target for model fitting, as wasusually used in earlier studies [Forget et al., 1998;Wood andPaige, 1992]. It should be noted, however, that the methodwe employ can easily include VL2 data simultaneously.[29] The Viking pressure transducers were calibrated to

about 2 Pa [Tillman, 1988]. This 2 Pa instrument error is theonly meaningful number that one can ascribe to ‘‘goodenough’’ in regards to fitting the observations. To fit betterthan 2 Pa could be unwarranted and to fit worse than 2 Pabetrays shortcomings in the model. Without some referenceto a meaningful numerical value, all fits are more or lessqualitative.[30] In order to fit the smoothed VL1 pressure cycle, we

use a simple linear method to first construct a best fitpressure curve from the baseline scenario plus a linearcombination of the 5 perturbation pressure vectors Pi. Wethen assume, as a consequence of linearity, the resulting bestfit MarsWRF parameters are equal to the baseline parame-ters plus the same linear combination of the perturbationparameter vectors, Di. If the linearity assumption holds,then we should find that nonlinearly validating the best fitparameters (by running MarsWRF with those values andapplying the interpolation and smoothing operators) yields apressure cycle very similar to both the best fit pressurecurve found from the linear combination of Pi, and thesmoothed VL1 measurements.[31] To illustrate the assumptions and consequences of

our linearity assumption, consider a vector of perturbations,dA, to the baseline parameter vector. If the nonlinear operatorF is applied to this perturbed parameter vector, then

X0 þ dX ¼ F A0 þ dAð Þ: ð15Þ

Performing a Taylor series expansion of the right hand sideof equation (15) about the baseline vector:

X0 þ dX ¼ F A0ð Þ þ FdAþ O k dA k2� �

; ð16Þ

where F is the Jacobian matrix of partial derivatives ofF(A0) with respect to the elements in A, evaluated about A0.Note that in our case F is a 669 � 5 matrix. If the vector ofparameter perturbations is small, where ‘‘small’’ is defined

by the ratio of the vector norms of dA and A0 being lessthan 1, then one can safely neglect the higher-order termsin equation (16), represented by O(kdAk2). Neglecting theseterms and subtracting the relation in equation (3) yields

dX ¼ FdA; ð17Þ

which is a linear relationship between perturbations inparameters and their resulting perturbations in the pressurecurve. An immediate consequence of this linear relation-ship is that a constant factor change in dA gives the samefactor change in dX.[32] The potential complexity of equation (17) is contained

within the matrix F (which itself is defined in equations (3)and (16)). While it is theoretically possible to evaluate theJacobian matrices of MarsWRF (W) with respect to specificparameters (i.e., the derivative of the model variables withrespect to the parameters) and also for the interpolation andsmoothing operators (I) for MarsWRF output, it is a difficultfeat requiring lots of code development. Instead of attemptingto explicitly calculate F, we approximate it by way of ourexplicit introduction of small perturbations to the parameterswithinA0 (as described in section 2). Hence, the columns ofF are related to the perturbation pressure vectors and themagnitudes of parameter perturbations:

Fi ¼ Pi= k Di k; i ¼ 1; 2; 3; 4; 5ð Þ; ð18Þ

where Fi are the columns of the F matrix. Alternatively,since the matrix with Di as its columns, which we denote D,is diagonal, then F = PD�1, where P is the perturbationmatrix with Pi as its columns and the superscript ‘‘�1’’denotes the matrix inverse.[33] To fit the smoothed VL1 data, we assume

Y ¼ X 0 þ Paþ e; ð19Þ

where Y denotes the smoothed VL1 data, a is a columnvector (5 � 1) of linear combination coefficients, and e is acolumn vector (669 � 1) of residuals. We find the best fitlinear combination coefficient via an iterative method thatminimizes the L2 norm (the root mean square) of e. Hence,we define a quadratic cost function that gauges the (possiblyweighted) magnitude of the residual vector:

C að Þ ¼ 0:5�eTWe ¼ 0:5� Y �X0� Pað ÞTW Y �X0� Pað Þ;ð20Þ

where W is a diagonal matrix that can be used to assignuneven weights to elements of the noise vector at dif-ferent times of the year (if desired). The cost function inequation (20) is of a standard form and can be easily min-imized by many different algorithms; because a has only 5or 7 elements here, we employ a straightforward downhillsimplex method. When the minimizing algorithm reaches a(possibly local) minimum for the cost function C, thecorresponding a = af minimizes e. The obtained best fitfor Y based on the linear combination is then

X f ¼ X0 þ Paf : ð21Þ


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[34] As the ultimate goal of this method is to find theMarsWRF parameter set that best fits the smoothed VL1record, we need to relate Xf in equation (21) to Af, the bestfit parameter set. If the linear truncation of the Taylor seriesin equation (16) holds, then changes in X can be linearlyrelated to changes in A, as in equation (17). Hence, weassume that the best fit parameter vector is related to thebaseline parameter vector by the same linear combinationcoefficients that relate the best fit pressure cycle to thebaseline pressure cycle prediction:

Af ¼ A0 þ Daf : ð22Þ

In order to gauge the success of the fitting method, and thusthe validity of the linearity assumption, one must runMarsWRF with parameters Af and apply the interpolationand smoothing operator to the output:

X 0f ¼ F Af

� �: ð23Þ

[35] If X0f is very similar to the linear method predic-

tion, Xf, then we can be assured the linearity assumptionwas valid. If X0

f and Xf differ substantially, then one mustreevaluate use of this method, though we have found thatfor poorly chosen baseline parameters, successive appli-cations of the method can eventually yield an acceptablesolution for Af .[36] When we write X = F(A), we consider the surface

pressure to be a function of the albedo and the emissivity ofthe polar caps and the total mass of CO2 in the system (theextended vector also considers the thermal inertia of thesubsurface ice table). By virtue of the linearity assumption,this method effectively retrieves these parameters Af, fromobservations of surface pressure, even though the involvedoperators (I and W) are nonlinear. In this respect, this linearfitting method is not new. The method is essentially a stan-dard practice in atmospheric spectroscopy data retrieval [e.g.,Guo et al., 2007] and data assimilation [e.g., Menemenlis etal., 2005].

4. Results

4.1. Standard Fitting With Emissivity, Albedo,and Total CO2 Mass

[37] We obtain a positive perturbation matrix, P+, toreplace P in equation (19):

Pþ ¼ P1 P2 P3 P4 P5½ �: ð24Þ

We describe it as ‘‘positive’’ because all the columns in P+

were obtained by setting one of the five parameters larger thanthat in the baseline case and differencing the resulting pressurecycles. Similarly, we can construct a negative perturbationmatrix P�, based on negative perturbations. We fit the VL1pressure record with these two perturbation matrices andobtain two best fit parameter vectors (also listed in Table 1):

Aþf 4:1 ¼ 0:796 0:484 0:467 0:787 0:978½ �T; ð25Þ

A�f 4:1 ¼ 0:793 0:485 0:454 0:784 0:978½ �T: ð26Þ

The values in these two parameter vectors are very close. Weaverage the two best fit parameter sets to get the average bestfit parameter vector:

Af 4:1 ¼ 0:795 0:485 0:461 0:785 0:978½ �T: ð27Þ

The two linear fits corresponding to Af4.1+ and Af4.1

� , andtheir average are shown in Figure 4 (top); their residualsare shown in Figure 4 (bottom). All fitted curves matchvery closely with the smoothed VL1 data. The largestmisfits are found in the northern winter, when baroclinicwaves are most active [Hess et al., 1977]. The largest erroris less than 10 Pa, or about 1% of the annual maxima.[38] We use the average parameter vector Af4.1 to drive

MarsWRF. Figure 5 shows the resulting model output. Thepressure cycles corrected for VL positions are indicated bythe black and the gray curves. The smoothed model surfacepressure cycles (green and magenta dashed curves) showgreat agreements with the smoothed VL data (red and bluedashed curves) at both landing sites (note that only theVL1 data were used for the fit). Figure 5 (bottom) showsthe difference between the smoothed simulation and thesmoothed data. For the VL1 site, the error is always lessthan several Pascal. The residual mean is 0.3 Pa and thestandard deviation is 3.2 Pa. The RMS of the residual is3.2 Pa, or 0.35–0.48% of the seasonal cycle. MarsWRFpredicts slightly higher surface pressure near Ls = 20� and240�, lower surface pressure near Ls = 150� and 270�.Similar, if not identical, residual patterns can be found inthe linear fit (blue dashed line in Figure 4, bottom). It sug-gests that our final perturbation to A0 is small enough forlinearity to hold (equation (19)). In this regime, the operatorF can be considered close to linear. The translation back tothe parameter space is therefore valid and reflected directlyin the forward model (i.e., MarsWRF) output.[39] We notice that the smoothed surface pressure cycle at

the VL2 site follows the observation closely in most of theyear, but it does not line up perfectly with the observationsin the northern winter and around Ls = 50�. Because wedo not perform the pressure fitting for the VL2 data, suchdiscrepancy is not surprising. Nonetheless, the fact that themodel agrees with data even at the VL2 site for the majorityof the year is impressive. It suggests that the hydrostaticassumption and the dynamics in MarsWRF are consistentwith what actually happens on Mars. If pressure records atboth VL sites are to be fit simultaneously, we would need toinclude the VL2 pressure responses in the perturbationmatrix and in the definition of the cost function. A trade-off of accuracy between the two landing locations is there-fore expectable. For the reasons discussed at the beginningof section 3, we decided to perform the fitting only for theVL1 pressure cycle.[40] If achieving a better fit for a desired time frame is the

goal, we can adjust the weighting assigned to differentperiods of time by modifying matrix W in the definition ofthe cost function (equation (20)). In Figures 4 and 5, one canfind a subtle phase error at the pressure minima near Ls =150�. In order to improve the fitting quality at this season, weweight this period relatively more in the calculation of thecost function. We chose to assign 50 times the normalweighting to southern winter (Ls = 125� to 175�), 10 times


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to southern summer (Ls = 225� to 275�) and normal weight-ing at all other times. With the updated weight matrix, thefollowing best fit parameter vectors are given by the linearfitting method (also shown in Table 1):

Aþf 4:2 ¼ 0:814 0:424 0:461 0:744 0:966½ �T; ð28Þ

A�f 4:2 ¼ 0:820 0:434 0:433 0:764 0:971½ �T: ð29Þ

We average the two to obtain

Af 4:2 ¼ 0:817 0:429 0:447 0:754 0:968½ �T: ð30Þ

This parameter vector provides an excellent linear fit aswell, with residual mean of 0.5 Pa and standard deviation of4.6 Pa. When validated using MarsWRF, the RMS of theresidual is 4.8 Pa, or 0.53–0.70% of the pressure level. Inboth linear fit and model output, the residual in the southern

winter (Ls = 125� to 175�) is smaller and the phase shift isless evident. The trade off, however, is that the errors atother seasons become larger, which leads to a larger totalRMS error.[41] The experiments above suggest that the southern

seasonal cap has low albedo and high emissivity, while thenorthern seasonal cap has high albedo and low emissivity.There is no intuitive explanation for why the two seasonalCO2 caps would have opposite radiative properties. Obser-vation from TES and Viking IRTM do not support such adichotomy in general [Kieffer et al., 1977; Kieffer and Titus,2001; Paige et al., 1994; Paige and Keegan, 1994]. With justthe fitting parameters used in this section, the southernparameters seem to match the observation from spacecraftsmuch better than those retrieved for the north.

4.2. Fitting With Extended Parameterizationof Subsurface Water Ice

[42] Since the conventional five parameter fit does notprovide seasonal cap albedos and emissivities consistent

Figure 4. The linear fit of surface pressure cycle at VL1 site. (top) Gray line, smoothed VL1 data; greendashed line, linear fit result using positive perturbation matrix P+; red dashed line, linear fit result usingnegative perturbation matrix P�; blue dashed line, the average of the previous two linear fits. (bottom)Green dashed line; residual in the fit using P+; red dashed line, residual in the fit using P�; blue dashedline, the average residual.

Table 1. Fitting Parameters for Different Perturbation Matrix and Time Weighting

Methods

Parameters

Northern CapAlbedo

Northern CapEmissivity

Southern CapAlbedo

Southern CapEmissivity

Total Mass of CO2

in Systema

Positive perturbation constant weighting (A4,1+ ) 0.796 0.484 0.467 0.787 0.978

Negative perturbation constant weighting (A4,1� ) 0.793 0.485 0.454 0.784 0.978

Average, constant weighting (Af4,1) 0.795 0.485 0.461 0.785 0.978Positive perturbation, nonconstant weighting (A4,2

+ ) 0.814 0.424 0.461 0.744 0.966Negative perturbation, nonconstant weighting (A4,2

� ) 0.820 0.434 0.433 0.764 0.971Average, nonconstant weighting (Af4,2) 0.817 0.429 0.447 0.754 0.968

aMass index number, 1.0 corresponds to total mass 2.90 � 1016 kg of the baseline case.


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with prior observations, especially in the north, we try toimprove the physics picture by including the subsurfacelayer thermal property as fitting parameters. As demonstratedin section 2.3, sensitivity studies show that the water icetable thermal inertia has a similar footprint to that of theemissivity. On the other hand, it does not project a changein the surface pressure cycle that is as ‘‘linear’’ as that ofemissivity. However, as long as the perturbation is not toolarge, we can still assume that the linear relationship holds.[43] First, we try to extend the parameter vector to seven

dimensions by adding the thermal inertia of the water icetable in the two hemispheres to the parameter vector. Weextend the baseline parameter by two more dimensions aswell and rewrite the baseline parameter vector as

A0 ¼ 0:770 0:570 0:500 0:800 1:00 0:275 0:275½ �T:

[44] The first five dimensions are defined as before. Thesixth and seventh dimensions correspond to the thermalinertia of the water ice table in the north and south. In thebaseline case, the subsurface is completely filled with drysoil. Therefore, the thermal inertia of the water ice table isassumed to be 275 J m�3 K�1, which is a value for averagedry Martian soil. The perturbation vector P is also extendedto

P ¼ P1 P2 P3 P4 P5 Pe6 P

e7

� �: ð31Þ

[45] Note that P6e and P7

e were calculated with a differentbaseline value of the water ice table thermal inertia. In orderto perform the linear fitting, we need to assume that thesetwo pressure perturbation vectors do not change regardless of

the baseline value. Undertaking a separate sensitivity study,we find that this is a reasonable assumption for the baselinevalues of 275 and 1200 J m�3 K�1, but later in this section wewill examine a case in which this assumption needs to bereevaluated. Therefore, we use the same linear fitting methodand obtain a new best fit parameter vector

Af 4:3 ¼ 0:790 0:505 0:474 0:788 0:977 0:523 0:300½ �T: ð32Þ

[46] This parameter vector has no essential difference towhat we obtain from the conventional five-dimensional fit(Af4.1). The corrections to the parameters are just a fewthousandths. When validated with MarsWRF, the residualpattern is very similar to before with slightly higher RMSerror (about 3.97 Pa). It suggests that including a water icetable thermal inertia as fitting parameters will not producea great improvement in the fitting. Thus, the conventionalfive-dimensional fit is sufficient for most atmosphericstudies, in which the important thing is to obtain a surfacepressure cycle that is within instrument error, while theaccuracy of the retrieved frost radiative properties areconsidered less relevant (and indeed, obviously incorrectvalues can be tolerated, on the assumption that nonmodeledphysics are aliased in these values).[47] As discussed by Haberle et al. [2008], the signifi-

cance of including subsurface ice is in a desire to gainphysical consistency, i.e., being able to fit the VL pressurecycles with reasonable cap properties close to their observedvalues. We setup an experiment using unity frost emissivity,relatively small frost albedos, and subsurface water ice withfixed uniform depths to the water ice table (Figure 2) asassumed by Haberle et al. [2008]. We find that we are ableto reproduce a pressure cycle reasonably close to that mea-

Figure 5. Same as Figure 1, except MarsWRF outputs are generated using the best fit parameters(northern seasonal cap albedo, 0.795; northern seasonal cap emissivity, 0.485; south albedo, 0.461; southemissivity, 0.785; total CO2 mass index, 0.978).


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sured by VL1 (Figure 6, consistent with prior ‘‘by-eye’’fitting but not as close as our five-parameter retrieval). Whilethe amplitude of the cycle agrees well with the observation,the major discrepancy between the simulation and data isin the first half of the year, where the phases of the two curvesdo not perfectly agree. The same mismatch in phase can befound in the NASA Ames Research Center (ARC) MarsGCM simulations.[48] The depth of the ice table will not in general be

uniform in latitude or longitude, yet its depth is evidentlyreadily predicted by simple 1-D water exchange models[Mellon and Jakosky, 1993; Schorghofer and Aharonson,2005]. Following the philosophy of trying to find the mostphysically reasonable fit parameters, we decided to employlatitudinally varying depths of the water ice table as derivedby Schorghofer and Aharonson [2005]. When we use thesedepths values and reduce the thermal inertia of the water icetables from 2200 J m�3 K�1, we are also able to reproducea reasonable fit to the VL pressure cycles (Figure 7). Thequality of the fit is similar to that using the aforementionedARC GCM setup, with the same phase mismatch during thenorthern summer. In this simulation, the water ice table topdepends on latitude, and the thermal inertia assumed for thewater ice table is not as high as pure water ice, with largervalues for the north. If we also assume the same mixingratio of water to dry soil in the water ice table, we find thatthe thermal inertia of the dry soil in the north is higher thanthe south, which agrees with IRTM and TES surface thermalinertia maps.[49] Our sensitivity study has already shown that the

seasonal cap emissivity and the water ice table thermalinertia have very similar footprints in the VL pressure cycle.

In other words, the role of emissivity in the linear fittingmay be essentially replaced by the water ice table thermalinertia (and by extension, the emissivity from the conven-tional five-parameter fit may include aliased effects fromneglected thermal inertia variations). Therefore, we proceedto do the numerical fitting with unity emissivity for theseasonal caps and instead vary the water ice table thermalinertia. In this case, the parameter vector still has fivedimensions but the two dimensions corresponding to theseasonal cap emissivity are replaced by the thermal inertiaof the water ice table in the north and south (their depthderived by Schorghofer and Aharonson [2005]). The best fitparameter vector thus retrieved is

Af 4:7 ¼ 0:731 0:546 0:948 3:90 1:80½ �T: ð33Þ

[50] The new fitting result is show in Figure 8. The RMSof the residual is now roughly 6.9 Pa. When we use theseparameters in MarsWRF, we are able to replicate the VL1record very well (shown in Figure 9). The RMS of the erroris 8.7 Pa, slightly larger than with the prior five-parameter(and seven-parameter) fit, but satisfactory. No obvious phasedisagreement between the two smoothed curves can befound.[51] In summary, we succeed in fitting the VL data with

the values of the albedos of the seasonal caps, the total CO2

inventory, and the subsurface layer thermal inertia, alongwithan assumption of unity seasonal cap emissivity. Although thefitting still retrieves an albedo of the northern seasonal caphigher than the south, the values are closer to the observation.The linear fitting results suggest that polar regolith with a

Figure 6. Same as Figure 1, except fitting parameter are the same as Haberle et al. [2008]. Northernseasonal cap albedo, 0.6; northern seasonal cap emissivity, 1.0; south albedo, 0.5; south emissivity, 1.0;total CO2 mass index, 1.003. Water ice table starts at 8.05 cm in the northern hemisphere and 11.16 cm inthe southern hemisphere. Thermal inertia of ice table is 2200 J m�3 K�1.


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higher thermal conductivity (due to subsurface water ice) isable to represent the extra heat source in autumn and winter,that used to be provided by excessively low cap emissivity instudies prior to Haberle et al. [2008]. The best fit thermalinertia values are very different in the two hemispheres,

seemingly necessary to account for the different pressurelevels during northern and southern winters. The predictedthermal inertia of the northern water ice table is much largerthan that of the south, suggesting some sort of dichotomy in

Figure 7. Same as Figure 1, except the driving parameters are as follows: northern seasonal cap albedo,0.6; northern seasonal cap emissivity, 1.0; south albedo, 0.5; south emissivity, 1.0; total CO2 mass index,1.003; thermal inertia of the ice table is 1800 J m�3 K�1 in the northern hemisphere and 900 J m�3 K�1

in the south. Boundary between the dry soil and the water ice table is set to the boundary of permanentlystable water ice suggested by Schorghofer and Aharonson [2005].

Figure 8. (top) Smoothed VL pressure cycle (gray line) and the fitted pressure curve (dash line) usingalbedo of the two seasonal caps, total CO2 inventory, and the thermal inertia of the water ice table at twodifferent hemispheres. (bottom) Residual (fitted curve minus VL data).


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the thermal properties between the two hemispheres. We willdiscuss this issue in section 5.

5. Discussions of Parameterization

5.1. Albedo and Emissivity

[52] Although albedo and emissivity may be model-dependent parameters, their physical interpretations are uni-versal. By definition, albedo describes the fraction of solarenergy reflected back to space while emissivity measures thefraction of energy in the blackbody curve for a surface ofgiven kinetic temperature that is released from that emitter. Ifwe first look at the retrievals without consideration ofsubsurface water ice, the five parameter linear fit suggeststhat the southern seasonal cap has lower albedo (0.461) andhigher emissivity (0.785) than the north (0.795 and 0.485,respectively). For these experiments, MarsWRF was not runin a mode to account for activities that change the effectivealbedo and emissivity (such as clouds, dust storms, etc.).Therefore, the best fit parameters not only characterize theseasonal caps but also provide hints as to other physicalprocesses implicitly important to MarsWRF in the polarregions.5.1.1. Southern Seasonal Cap[53] When surface CO2 ice is mixed with dust, its albedo

and emissivity change significantly. Assuming the samegrain size of ice, a dustier deposit can produce a much lessreflective and much more emissive surface material [Kiefferet al., 2000]. Hence, we expect that we can attribute many ofthe albedo and emissivity characteristics to the dust activity,which is usually stronger in the southern hemisphere [Basu etal., 2004; Martin, 1986; Smith, 2004]. However, it is lessclear that increases in dust result in a generally darker andmore emissive cap [James et al., 2000].

[54] Isotropic albedo (usually slightly higher than Lam-bert albedo) was mapped by TES [Kieffer et al., 2000]. Thesouthern seasonal cap in general had albedos of 0.45 to 0.5,with lower values (about 0.3) in the so-called ‘‘Crypticregions’’ and higher values (0.6 to 0.7) in the bright cap. Onaverage, TES observations agree very well with the albedopredicted by the linear fit model for the south pole. Lambertalbedo was also mapped by IRTM aboard the Viking orbiters.It ranged from 0.2 to 0.5 in the south [Paige and Keegan,1994]. The region with an albedo 0.5 corresponded to theresidual cap rather than the seasonal cap. Since the southernresidual cap is believed to consist of mostly CO2 ice [Paigeand Ingersoll, 1985], the IRTM observations for the CO2 capare also consistent with our linear fit prediction. The HubbleSpace Telescope has reported higher southern seasonal capalbedos for the bright cap region [James et al., 2005].[55] We did not account for atmospheric CO2 cloud for-

mation and precipitation with microphysical calculations inthis version ofMarsWRF (when the atmospheric temperaturefalls below the saturation point temperature for CO2 conden-sation, the model instantaneously deposits the correspondingcondensed CO2 ice on the ground). It is believed thatatmospheric CO2 condensation may affect the atmosphericstate [Colaprete et al., 2008] and CO2 snowfall could con-tribute to the low brightness temperature or equivalently thelow emissivity observed by various missions [Colaprete etal., 2005; Forget et al., 1999]. Some models have madeefforts to relate the change in surface (seasonal cap) emis-sivity to the amount of condensed CO2. Forget et al. [1998]uses an empirical function to adjust the seasonal cap emis-sivity based on the atmospheric CO2 condensation andprecipitation rate. Using equation (5) of that paper, we cancalculate an average adjustment to the seasonal cap emissiv-ity due to CO2 snow. For the southern cap, this correction is

Figure 9. Same as Figure 7, except the driving parameters are as follows: northern seasonal cap albedo,0.731; northern seasonal cap emissivity, 1.0; south albedo, 0.546; south emissivity, 1.0; total CO2 massindex, 1.077; thermal inertia of the ice table is 3900 J m�3 K�1 in the northern hemisphere and 1800 J m�3

K�1 in the south.


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about 0.15. When subtracted from the reference value of0.95, the effective emissivity is about 0.8, which is very closeto our best fit value. In other words, if CO2 snow is important,our time-constant parameterization may be effectively cap-turing this physical phenomenon while avoiding complexityin computation.5.1.2. Northern Seasonal Cap[56] Measurements of the northern seasonal cap albedo

made by TES [Kieffer and Titus, 2001] are lower than ourbest fit value, and so are the IRTM observations [Paige etal., 1994; Paige and Keegan, 1994]. The polar hood cloud,which is usually seen in the winter time [James et al., 1992,1994; Wang and Ingersoll, 2002], could contribute to theexcessively high retrieved albedo of the northern cap. Fur-thermore, polar hood clouds are seen much less often in thesouthern hemisphere, and the retrieval results show thatlittle correction is needed for the albedo of the southern cap,which starts relatively close to the observations. Althoughthe radiative effects of hood clouds are not modeled byMarsWRF, the five-parameter set predicts the atmosphericmass budget for Mars very well. If hood cloud effects aresignificant, the retrieval scheme is effectively capturing(and aliasing) these effects into the northern seasonal capretrieved albedo, by increasing the best fit albedo by 0.3 to0.4 relative to the observations.[57] When the effect of dust on the surface CO2 ice cap is

ignored, emissivity can be considered as a function of theporosity of the CO2 frost: more porous ice tends to have loweremissivity [Eluszkiewicz et al., 2005]. Our best fit emissivityin the north is much lower than that in the south, suggesting amuch more porous CO2 ice slab in the north. Porosity data ofthe two seasonal caps, which would provide a direct evalu-

ation of our argument, are not yet available. However, ourargument is at least consistent with the extremely low densityof the northern seasonal cap [Aharonson et al., 2004].[58] Finally, it should be noted that a good fit to the VL

pressure data can still be obtained when the northern capemissivity is assumed to be unity as long as subsurfacewater ice effects are included. Thus the degree to whichthe low northern emissivities must be explained is itselfquestionable.

5.2. CO2 Inventory

5.2.1. Annual Variation[59] The global CO2 inventory cycle is plotted in Figure 10,

divided into different reservoirs. Our best fit cases suggestthat the total CO2 in the surface-atmosphere system is about2.83� 1016 kg, which agrees towithin 5% of theNASAARCGCM [Kelly et al., 2006]. 26% of the total CO2 (7.45 �1015 kg) participates in the seasonal exchange betweenthe atmosphere and the caps, slightly larger than the ARCGCM predictions, but consistent with earlier estimation[James et al., 1992; Kelly et al., 2006].[60] As discussed in the sensitivity study, the amount of

surface CO2 ice in MarsWRF is mainly determined by thealbedo and emissivity of the corresponding seasonal caps.The prediction of surface CO2 in the southern hemisphereby MarsWRF matches closely with the GRS observations.In the north, our prediction is slightly higher than GRSobservations [Kelly et al., 2006]. Ideally, GCMs should beable to simulate both surface CO2 deposition and VL sitepressure cycles that agree well with the available observa-tions, thus yielding an estimate of the total ‘‘active’’ CO2

budget in the system. Slight disagreements in simultaneously

Figure 10. CO2 budget in MarsWRF using the best fit parameters (northern seasonal cap albedo, 0.795;northern seasonal cap emissivity, 0.485; south albedo, 0.461; south emissivity, 0.785; total CO2 massindex, 0.978). Red line, total CO2 in the system; blue line, mass of CO2 in the atmosphere; black line,mass of CO2 on the surface; magenta line, surface CO2 in the southern hemisphere; green line, surfaceCO2 in the northern hemisphere.


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fitting the seasonal cap masses and lander pressure curvescould result from interpolation errors given the relativelypoor model resolution (5� � 5.625�), missing sources andsinks of CO2 (regolith), or instrument error.5.2.2. Permanent Polar Caps[61] A permanent CO2 cap exists at the Martian southern

pole [Kieffer et al., 1977]. No published GCM predicts theexistence of such a residual cap, and instead a residual capmust be prescribed in the model boundary conditions (oftensimply as a fixed surface temperature at the CO2 frostpoint). We have found that permanent CO2 caps will appearin MarsWRF only if extreme parameters (e.g., very highalbedo values) are used, which agree withWood and Paige[1992].[62] The best fit parameter sets, which lead to good rep-

lications of the VL pressure cycle, do not produce perma-nent CO2 caps. This is not unexpected. Other GCMs do notproduce permanent caps while trying to fit the VL pressurecycles [Haberle et al., 2008]. As stated in the introduction,Wood and Paige [1992] used a simple one dimension modelfor the thermal calculation to find the best parameter set tofit the VL pressure cycle. They pointed out that with theirbest fit values, the permanent cap is absent in both poles.We verified their conclusion once again using a much moresophisticated model (a GCM) that something ‘‘special’’ ishappening in the heat balance at the south residual cap tosomehow allow its existence. This is possibly due to aninsolation-dependent albedo [James et al., 1992; X. Guo etal., On the mystery of the perennial carbon dioxide cap atthe south pole of Mars, submitted to Journal of GeophysicalResearch, 2009].

5.3. Subsurface Water Ice

[63] As discussed in the sensitivity study, most of thesubsurface water ice layer’s footprint on the VL pressurecycle can be represented by a linear combination of theseasonal cap albedo and emissivity. Fitting using subsurfacewater ice (via the effects on subsurface conductivity) doesnot give particularly better fits. Rather, the advantage offitting using surface ice is in retrieving cap properties thatare more physically plausible. Subsurface water ice con-tributes directly to the seasonal energy balance, which is thedriver of the CO2 annual cycle. Larger thermal conductivityprovides the missing heat source in the energy balance inlocal winter, which before was represented by unrealisticfrost albedo and emissivity. Its potential was already dem-onstrated in the GCM study by Haberle et al. [2008].[64] The biggest difficulty in completing the picture is

that we do not know precisely how deep the water ice isburied, what the water content is in the ice table, and howthings change over time. The current assumptions inMarsWRF and ARC GCMs are more or less arbitrary. Theyare not backed up bymany systematic observations, althoughthe Phoenix lander observations do support the kind of icetable depth modeling undertaken by Mellon and Jakosky[1993] and Schorghofer and Aharonson [2005]. On the otherhand, given very accurate measurements of the albedo andemissivity of the seasonal caps, we would likely be able touse the technique we have described to retrieve the (bulk)properties of subsurface water ice.[65] The fit with CO2 albedo, total CO2 mass, and water

ice table thermal inertia suggests much higher thermal

conductivity for the subsurface layer in the north, to at leastseveral meters below the surface. Since the spatial coverageof the water ice table is very different in the two hemi-spheres according to the GRS measurements (Figure 2), thedifference in the conductive properties per unit mass of theregolith must be even larger. This is potentially a constrainton the nature of the Martian near-surface regolith. Thecrusts of the two hemispheres have long been known todiffer [Neumann et al., 2004;Watters et al., 2007], so it maynot be surprising to find they have different thermalconductive properties (or different abilities to accommodatehighly conductive water ice).[66] GRS reported hydrogen content and equivalent water

content distributions (Figure 2) near the Martian surface[Feldman et al., 2004]. Assuming water ice occupies all thepores in the soil, and knowing the densities of water ice(900 kg m�3) and Martian crust (2900 kg m�3) [Zuber etal., 2000], we can back out the porosity of the soil. Wemake two more assumptions: first, the subsurface maintainsthe same porosity through the vertical column consideredand that the pores are completely filled with water ice;second, the dry soil thermal conductivity in the subsurfaceis the same as the surface, whose equivalent thermal inertiais provided by TES [Putzig et al., 2005]. We can then cal-culate the thermal inertia of the subsurface mixture of soiland water ice (shown in Figure 11). An obvious north-southdichotomy in thermal inertia map can be seen. Zonalaverage subsurface thermal inertia ranges from 300 to1400 J m�3 K�1 in the north, with a maximum of 1730 J m�3

K�1. In the south, the zonal average varies between 220 and540 J m�3 K�1; this is on average about two to three timessmaller than in the north. This dichotomy is due to thecombined effects of high water ice content and high soilthermal inertia in the northern polar region. We notice thatthe values of thermal inertia obtained from this calculationare smaller than suggested by the retrieval scheme. Thereare ways to reduce this gap since the best fitted thermalinertia values could be misleading: MarsWRF assumesconstant density and thermal capacity for the subsurfacematerial. A decrease in dry soil density and/or heat capacitywill decrease the thermal inertia while keeping the samethermal conductivity. One may also argue that the GRSmayunderpredict the water ice content and therefore the soilporosity. Setting those possibilities aside, there is an impor-tant common message from these two independent studies;that is, there exists a significant north-south dichotomy in thesubsurface thermal conductivity in the polar regions.[67] When we fix the seasonal cap emissivity to unity, we

still need to adjust the seasonal cap albedos in order to makethe seasonal pressure cycle phase match observations. Thissuggests that the inclusion of the water ice table, whileimproving physicality, is not the panacea - we do not yethave a complete and satisfactory understanding of theprocesses controlling the CO2 cycle. In order to build abetter Mars GCM, we need to build more physically basedprognostic models for the frost albedo and emissivity,capturing processes now implicitly aliased into our retrievedcap properties.

5.4. Limitations of the Linear Fitting Method

[68] As we discussed in section 3, the linear fittingalgorithm assumes the response in surface pressure scales


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linearly with the perturbation of the fitting parameters. InMarsWRF, this assumption becomes less valid when theperturbation is large, especially for perturbations to the waterice table thermal inertia. For example, in Figure 3 (bottom),when we perturb the thermal inertia baseline by 1000, 2000,and 3000 J m�3 K�1, the responses of the latter two are notsimple multiples of the first. The perturbation matrix maychange greatly when the baseline parameter vector moves toa different state. It may influence the calculation of the costfunction and the iterativemethod: the last case from section 4.2provides a good demonstration of these effects. If this is thecase, we need to recalculate the pressure perturbation vectorsto maintain linearity.[69] The pattern of the perturbation matrix changes dras-

tically in extreme situations. More specifically, MarsWRFdoes not respond to further change of parameters after criticalvalues are reached. For instance, in Figure 3 (bottom), raisingthe water ice table thermal inertia by 3000 J m�3 K�1 doesnot increase the surface pressure much more than raising it by2000 Jm�3 K�1. In another situation where the frost albedo istoo high, a permanent cap will form in the correspondingpole. One can easily imagine that the pressure responsewould have a very different pattern.[70] Another related limitation of the iterative algorithm

is that it is sensitive to the choice of the starting parametervector. Starting the fitting with Y1 = F(A1) may produce adifferent best fit parameter vector than starting with Y2 =F(A2), but both may yield comparable RMS errors. Whenused in MarsWRF, however, they may correspond to verydifferent VL pressure cycles and RMS errors. Some are closerto the observations than the others. We have to choose thedesired sets on the basis of their performances in the GCMand their physical soundness.

[71] Ideally, we should force the perturbations to be verysmall and recalculate the perturbation matrix for eachiterative step. However, this wouldmake the fitting extremelyslow owing to the greater number of required iterations and,more importantly, more GCM runs to calculate the perturba-tion matrices. Fortunately, we can assume the perturbationmatrix does not change for most of our retrievals. We alsocarefully selected the starting point of the iteration to makethe process stable.

6. Conclusion

[72] We provide a method to obtain close fits to the VLsurface pressure data using a chosen subset of ‘‘tunable’’parameters within Mars General Circulation Models. Weconstruct a set of basis vectors by perturbing five parame-ters in the GCM, including albedo and emissivity for theseasonal CO2 caps in both poles and the total amount ofCO2 in the system. In this sense, our method is related toensemble data assimilation. We utilize an iterative methodto find the best linear combination of the basis vectors thatminimizes the least squares error between the fit and the VLdata. The coefficients for the linear fit are then projectedback to parameter space and a best fit parameter set isobtained. When used in MarsWRF, this parameter set yieldsa pressure cycle very close to the VL observations aspredicted by the linear method.[73] The method described in this paper provides a first

rigorous and quantitative means for fitting the seasonalpressure cycle with a GCM. As designed, we expect thatthis method would be useful for calibrating the CO2 cycle inany model and thus could provide a benchmark method fortuning GCMs. The study confirms the idea that a simple

Figure 11. (left) Hypothesized subsurface thermal inertia in J m�3 K�1 map and (right) zonal average.


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parameterization has the potential to capture most of theunderlying physics of the CO2 cycle [Wood and Paige,1992]. In the linear fit, the best fit parameters qualitativelyshow footprints of the physical processes that are notexplicitly represented in MarsWRF but effectively capturedby the parameterization, such as the presence of cloud,precipitation of atmospheric CO2 ice, surface frost withdifferent porosity, etc.[74] While the mathematics of the approach is elegant, the

retrieved parameters are not guaranteed to be in agreementwith direct observations. For instance, the cause of the bigdifference in the retrieved radiative properties of the twoseasonal caps is not thoroughly understood. Attributing thedifferences to specific unmodeled physical process willrequire further work, along the philosophical lines laid outin the Introduction. However, our method provides a roadmap for the procedure of unveiling all relevant physics. Westart with the simplest parameter set that fits the data well.Wherever the retrieved parameters differ from observed orphysically sound values, we must look for further unre-solved (unmodeled) physical processes. For example,parameterizations of clouds, dirty frost, etc., are likely stillrequired. The soil thermal property change due to thepresence of subsurface water ice is one such piece ofphysics we added in this study. Both MarsWRF and theNASA ARC Mars GCM have verified that including thesubsurface ice improves the physical soundness of retrievedcap properties.[75] Our simulations suggest that the northern subsurface

layer has higher thermal conductivity than the south. Thisargument is also supported by GRS and TES observations.This bias in the inferred thermal conductivity may be relatedto differences in the water cycle, in the ability of the regolithmaterials in the two hemispheres to accommodate water,and/or in the regolith conductivity itself. Finally, none of theretrievals reproducing reasonable pressure curves are able tosimultaneously yield a residual CO2 ice cap at the south.This may suggest that assumptions such as the temporallyconstant values of the retrieved cap properties may not becompletely valid.

[76] Acknowledgments. We thank Oded Aharonson for the surfacewater content data, permanently stable water ice depth, and useful sugges-tions. We thank Andrew Ingersoll, Ashwin Vasavada, Stephen Wood, andYuk Yung for helpful discussions. We thank two anonymous reviewers fortheir insightful comments. The numerical simulations for this research wereperformed on Caltech’s Division of Geological and Planetary Sciences Dellcluster (CITerra).

ReferencesAharonson, O., M. T. Zuber, D. E. Smith, G. A. Neumann, W. C. Feldman,and T. H. Prettyman (2004), Depth, distribution, and density of CO2

deposition on Mars, J. Geophys. Res., 109, E05004, doi:10.1029/2003JE002223.

Basu, S., M. I. Richardson, and R. J. Wilson (2004), Simulation of theMartian dust cycle with the GFDL Mars GCM, J. Geophys. Res., 109,E11006, doi:10.1029/2004JE002243.

Boynton, W. V., et al. (2002), Distribution of hydrogen in the near sur-face of Mars: Evidence for subsurface ice deposits, Science, 297(5578),81– 85, doi:10.1126/science.1073722.

Colaprete, A., J. R. Barnes, R.M. Haberle, J. L. Hollingsworth, H. H. Kieffer,and T. N. Titus (2005), Albedo of the south pole on Mars determined bytopographic forcing of atmosphere dynamics, Nature, 435, 184–188,doi:10.1038/nature03561.

Colaprete, A., J. R. Barnes, R. M. Haberle, and F. Montmessin (2008), CO2

clouds, CAPE and convection on Mars: Observations and general circu-lation modeling, Planet. Space Sci., 56(2), 150–180.

Eluszkiewicz, J., J. L. Moncet, T. N. Titus, and G. B. Hansen (2005), Amicrophysically based approach to modeling emissivity and albedo of theMartian seasonal caps, Icarus, 174(2), 524 – 534, doi:10.1016/j.icarus.2004.05.025.

Feldman, W. C., et al. (2002), Global distribution of neutrons from Mars:Results from Mars Odyssey, Science, 297(5578), 75–78, doi:10.1126/science.1073541.

Feldman, W. C., et al. (2004), Global distribution of near-surface hydrogenon Mars, J. Geophys. Res., 109, E09006, doi:10.1029/2003JE002160.

Forget, F., F. Hourdin, and O. Talagrand (1998), CO2 snowfall on Mars:Simulation with a general circulation model, Icarus, 131(2), 302–316,doi:10.1006/icar.1997.5874.

Forget, F., F. Hourdin, R. Fournier, C. Hourdin, O. Talagrand, M. Collins,S. R. Lewis, P. L. Read, and J. P. Huot (1999), Improved general circula-tion models of the Martian atmosphere from the surface to above 80 km,J. Geophys. Res. , 104(E10), 24,155 – 24,175, doi:10.1029/1999JE001025.

Guo, X., V. Natraj, D. R. Feldman, R. J. D. Spurr, R. Shia, S. P. Sander, andY. Yung (2007), Retrieval of ozone profile from ground-based measure-ments with polarization: A synthetic study, J. Quant. Spectrosc. Radiat.Transfer, 103(1), 175–192, doi:10.1016/j.jqsrt.2006.05.008.

Haberle, R. M., F. Forget, A. Colaprete, J. Schaeffer, W. V. Boynton, N. J.Kelly, and M. J. Chamberlain (2008), The effect of ground ice on theMartian seasonal CO2 cycle, Planet. Space Sci., 56(2), 251 – 255,doi:10.1016/j.pss.2007.08.006.

Hess, S. L., R. M. Henry, C. B. Leovy, J. A. Ryan, and J. E. Tillman (1977),Meteorological results from the surface of Mars: Viking 1 and 2,J. Geophys. Res., 82(28), 4559–4574, doi:10.1029/JS082i028p04559.

Hourdin, F., P. Levan, F. Forget, and O. Talagrand (1993), Meteorologicalvariability and the annual surface pressure cycle on Mars, J. Atmos. Sci.,50(21), 3625–3640, doi:10.1175/1520-0469(1993)050<3625:MVATAS>2.0.CO;2.

James, P. B., H. H. Kieffer, and D. A. Paige (1992), The seasonal cycle ofcarbon dioxide on Mars, in Mars, edited by H. H. Kieffer et al., pp. 934–968, Univ. of Ariz. Press, Tucson.

James, P. B., R. T. Clancy, S. W. Lee, L. J. Martin, R. B. Singer, E. Smith,R. A. Kahn, and R. W. Zurek (1994), Monitoring Mars with the Hubble-Space-Telescope: 1990–1991 Observations, Icarus, 109(1), 79 –101,doi:10.1006/icar.1994.1078.

James, P. B., et al. (2000), The 1997 spring regression of the Martian southpolar cap: Mars Orbiter camera observations, Icarus, 144(2), 410–418,doi:10.1006/icar.1999.6289.

James, P. B., B. P. Bonev, and M. J. Wolff (2005), Visible albedo of Mars’south polar cap: 2003 HST observations, Icarus, 174, 2, 596–599,doi:10.1016/j.icarus.2004.08.024.

Kelly, N. J., W. V. Boynton, K. Kerry, D. Hamara, D. Janes, R. C. Reedy,K. J. Kim, and R. M. Haberle (2006), Seasonal polar carbon dioxidefrost on Mars: CO2 mass and columnar thickness distribution, J. Geophys.Res., 111, E03S07, doi:10.1029/2006JE002678.

Kieffer, H. H., and T. N. Titus (2001), TES mapping of Mars’ north sea-sonal cap, Icarus, 154(1), 162–180, doi:10.1006/icar.2001.6670.

Kieffer, H. H., T. Z. Martin, A. R. Peterfreund, B. M. Jakosky, E. D. Miner,and F. D. Palluconi (1977), Thermal and albedo mapping of Mars duringthe Viking primary mission, J. Geophys. Res., 82(28), 4249–4291,doi:10.1029/JS082i028p04249.

Kieffer, H. H., T. N. Titus, K. F. Mullins, and P. R. Christensen (2000),Mars south polar spring and summer behavior observed by TES: Seasonalcap evolution controlled by frost grain size, J. Geophys. Res., 105(E4),9653–9699, doi:10.1029/1999JE001136.

Martin, T. Z. (1986), Thermal infrared opacity of the Mars atmosphere,Icarus, 66(1), 2–21, doi:10.1016/0019-1035(86)90003-5.

Mellon, M. T., and B. M. Jakosky (1993), Geographic variations in thethermal and diffusive stablity of ground ice on Mars, J. Geophys. Res.,98(E2), 3345–3364.

Menemenlis, D., I. Fukumori, and T. Lee (2005), Using Green’s functionsto calibrate and ocean general circulation model,Mon. Weather Rev., 133,1224–1240, doi:10.1175/MWR2912.1.

Mischna, M. A., J. F. Kasting, A. Pavlov, and R. Freedman (2000), Influ-ence of carbon dioxide clouds on early Martian climate, Icarus, 145(2),546–554, doi:10.1006/icar.2000.6380.

Neumann, G. A., M. T. Zuber, M. A. Wieczorek, P. J. McGovern, F. G.Lemoine, and D. E. Smith (2004), Crustal structure of Mars from gravityand topography, J. Geophys. Res., 109, E08002, doi:10.1029/2004JE002262.

Owen, T., K. Biemann, D. R. Rushneck, J. E. Biller, D. W. Howarth, andA. L. Lafleur (1977), The composition of the atmosphere at the surfaceof Mars, J. Geophys. Res., 82(28), 4635 – 4639, doi:10.1029/JS082i028p04635.

Paige, D. A. (1992), The thermal stability of near-surface ground ice onMars, Nature, 356, 43–45, doi:10.1038/356043a0.


18 of 19

E07006

Paige, D. A., and A. P. Ingersoll (1985), Annual heat balance of the Martianpolar caps: Viking observations, Science, 228(4704), 1160 – 1168,doi:10.1126/science.228.4704.1160.

Paige, D. A., and K. D. Keegan (1994), Thermal and albedo mapping ofthe polar regions of Mars using Viking thermal mapper observations:2. South polar region, J. Geophys. Res., 99(E12), 25,993 –26,013,doi:10.1029/93JE03429.

Paige, D. A., J. E. Bachman, and K. D. Keegan (1994), Thermal and albedomapping of the polar regions of Mars using Viking thermal mapperobservations: 1. North polar region, J. Geophys. Res., 99(E12),25,959–25,991, doi:10.1029/93JE03428.

Pollack, J. B., R. M. Haberle, J. R. Murphy, J. Schaeffer, and H. Lee(1993), Simulations of the general-circulation of the Martian atmosphere:2. Seasonal pressure variations, J. Geophys. Res., 98(E2), 3149–3181,doi:10.1029/92JE02947.

Putzig, N. E., M. T. Mellon, K. A. Kretke, and R. E. Arvidson (2005),Global thermal inertia and surface properties of Mars from the MGSmapping mission, Icarus, 173(2), 325 – 341, doi:10.1016/j.icarus.2004.08.017.

Richardson, M. I., and R. J. Wilson (2002), A topographically forcedasymmetry in the Martian circulation and climate, Nature, 416, 298–301, doi:10.1038/416298a.

Richardson, M. I., R. J. Wilson, and A. V. Rodin (2002), Water ice clouds inthe Martian atmosphere: General circulation model experiments with asimple cloud scheme, J. Geophys. Res., 107(E9), 5064, doi:10.1029/2001JE001804.

Richardson, M. I., A. D. Toigo, and C. E. Newman (2007), PlanetWRF:A general purpose, local to global numerical model for planetary atmo-spheric and climate dynamics, J. Geophys. Res., 112, E09001,doi:10.1029/2006JE002825.

Schorghofer, N., and O. Aharonson (2005), Stability and exchange of sub-surface ice on Mars, J. Geophys. Res., 110, E05003, doi:10.1029/2004JE002350.

Smith, M. D. (2002), The annual cycle of water vapor on Mars as observedby the Thermal Emission Spectrometer, J. Geophys. Res., 107(E11),5115, doi:10.1029/2001JE001522.

Smith, M. D. (2004), Interannual variability in TES atmospheric observa-tions of Mars during 1999–2003, Icarus, 167(1), 148–165, doi:10.1016/j.icarus.2003.09.010.

Tillman, J. E. (1988), Mars global atmospheric oscillations: Annually syn-chronized, transient normal mode oscillations and the triggering of globaldust storms, J. Geophys. Res., 93(D8), 9433 – 9451, doi:10.1029/JD093iD08p09433.

Tillman, J. E., N. C. Johnson, P. Guttorp, and D. B. Percival (1993), TheMartian annual atmospheric pressure cycle: Years without great duststorms, J. Geophys. Res., 98(E6), 10,963 – 10,971, doi:10.1029/93JE01084.

Wang, H., and A. P. Ingersoll (2002), Martian clouds observation by MarsGlobal Surveyor Mars Orbiter Camera, J. Geophys. Res., 107(E10),5078, doi:10.1029/2001JE001815.

Watters, T. R., P. J. McGovern, and R. P. Irwin III (2007), Hemispheresapart: The crustal dichotomy on Mars, Annu. Rev. Earth Planet. Sci., 35,621–652, doi:10.1146/annurev.earth.35.031306.140220.

Wood, S. E., and D. A. Paige (1992), Modeling the Martian seasonal CO2

cycle. 1. Fitting the Viking lander pressure curves, Icarus, 99(1), 1–14,doi:10.1016/0019-1035(92)90166-5.

Zuber, M. T., et al. (2000), Internal structure and early thermal evolution ofMars from Mars Global Surveyor topography and gravity, Science,287(5459), 1788–1793, doi:10.1126/science.287.5459.1788.

��X. Guo, W. G. Lawson, and M. I. Richardson, Division of Geological and

Planetary Sciences, California Institute of Technology, 1200 East CaliforniaBoulevard, MC150-21, Pasadena, CA 91125, USA. ([email protected])A. Toigo, Center for Radiophysics and Space Research, Cornell

University, Ithaca, NY 14850, USA.


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